Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

Yige Zhao; Yibing Sun; Zhi Liu; Yilin Wang; Yige Zhao; Yibing Sun; Zhi Liu; Yilin Wang

doi:10.3934/math.2020037

AIMS Mathematics

2020, Volume 5, Issue 1: 557-567. doi: 10.3934/math.2020037

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Research article

Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

1.
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P. R. China
2.
School of Automation and Electrical Engineering, Linyi University, Linyi, Shandong 276005, P. R. China
3.
Business School, University of Jinan, Jinan, Shandong 250002, P. R. China

Received: 10 October 2019 Accepted: 06 December 2019 Published: 12 December 2019
MSC : 34N05, 34A12

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Carathéodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.
- existence,
- boundary value problem,
- fractional differential equation,
- mixed perturbations
Citation: Yige Zhao, Yibing Sun, Zhi Liu, Yilin Wang. Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type[J]. AIMS Mathematics, 2020, 5(1): 557-567. doi: 10.3934/math.2020037

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Abstract

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Carathéodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

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